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User blog:Rgetar/Description of format for Veblen-like functions
Format for arrays Array is sequence of pairs "coordinates - element at this coordinates", separated with separators. Ordinals may be considered as arrays of natural numbers: "coordinates - element" pair is ωce, where c is coordinates, which is also ordinal, and e is element separator is "+" Ordinals are arrays of order 0. Array of order 1 is array of ordinals, and its coordinates are also arrays of order 1: "coordinates - element" pair is Ωce separator is "+" Array of order 2 is array of arrays of order 1, and its coordinates are also arrays of order 2: "coordinates - element" pair is Ω2ce separator is "+" Generally, array of order X is also array of all orders larger than X. And array of order X is array of arrays of orders lesser than X, and its coordinates are also arrays of order X: "coordinates - element" pair is ΩXce separator is "+" Ω0 = ω Ω1 = Ω Veblen-like functions φoutin(X) - this is Veblen-like function. Its input X is array of order in, and its output is array of order out. So, in is input order, and out is output order. Earlier I placed input order as subscript, and output order as superscript, but then I swaped them for several reasons, including my "rule" "strong-first": upper elements are first, and output is "stronger" than input. If in = 1, out = 0, then φ01(X) is generalized Veblen function. For array of order α X φαα + 1(X) = ΩαX So, Ωα can be expressed using φ: Ωα = φαα + 1(1) = φαα + 2(0) And φαα + 1(0) = 1 for any α. For array of order α + 1 X φαα + 1(X) is defined similarly to Veblen function. For example, φ12(Ω2) is limit of Ω, ΩΩ, ΩΩΩ, ... φ12(Ω2 + 1) is limit of φ12(Ω2), Ωφ12(Ω2) + 1, ΩΩφ12(Ω2) + 1, ... φ12(Ω22) is limit of φ12(Ω2), φ12(Ω2 + φ12(Ω2)), φ12(Ω2 + φ12(Ω2 + φ12(Ω2))), ... etc. Generally, φαα + 1(X) is in fact Veblen function with ω replaced by Ωα. Definition for out ≠ in + 1: φαβ(X) = φαγ(φγβ(X)) (By the way, the real reason why I swaped in and out is because φαβ(X) = φαγ(φγβ(X)) looks better than φβα(X) = φγα(φβγ(X))) Rules of omitting. Omitting subscript: φα(X) = φαα + 1(X) Omitting superscript: φα(X) = φ0α(X) Omitting subscript and superscript: φ(X) = φ01(X) - Veblen function Omitting φ: φαβ(X) = αβ(X) φα(X) = α(X) φβ(X) = β(X) φ(X) = (X) (But if we use φ(X) = (X), then we should not use (X) as array of more than one element to not be confused with Veblen function. For example, not Ω(ω + 1), only Ωω + Ω, because Ω(ω + 1) = Ωφ(ω + 1) = Ωωω + 1). Computer format "Computer format" for Veblen-like functions: φoutin(X)e = ⟨out.in.X⟩e For example, ⟨α.β.X⟩ = ⟨α.γ.⟨γ.β.X⟩⟩ ⟨α..X⟩ = ⟨α.α + 1.X⟩ ⟨α.X⟩ = ⟨0.α.X⟩ ⟨X⟩ = ⟨0.1.X⟩ Exploring ordinals This list is made by hand and based mainly on intuition. Currently I still have not strict definition of Veblen-like functions and fundamental sequence system for ordinals beyond BHO. FS #1 0 φ01(0) = φ12(0) = φ23(0) = φ34(0) = φ45(0) = ... = 1 2 3 4 5 6 7 ... FS #2 φ02(0) = φ01(φ12(0)) = φ01(1) = φ01(φ01(0)) = ω ω + 1 ω + 2 ω + 3 ω + 4 ω + 5 ω + 6 ω + 7 ... FS #3 φ02(0)2 = ω2 ω3 ω4 ω5 ω6 ω7 ω8 ... FS #4 φ01(2) = ω2 φ01(3) = ω3 φ01(4) = ω4 φ01(5) = ω5 φ01(6) = ω6 φ01(7) = ω7 φ01(8) = ω8 ... FS #5 φ01(φ01(1)) = φ01(φ01(φ01(0))) = ωω φ01(φ01(φ01(1))) = φ01(φ01(φ01(φ01(0)))) = ωωω φ01(φ01(φ01(φ01(1)))) = φ01(φ01(φ01(φ01(φ01(0))))) = ωωωω φ01(φ01(φ01(φ01(φ01(1))))) = φ01(φ01(φ01(φ01(φ01(φ01(0)))))) = ωωωωω φ01(φ01(φ01(φ01(φ01(φ01(1)))))) = φ01(φ01(φ01(φ01(φ01(φ01(φ01(0))))))) = ωωωωωω φ01(φ01(φ01(φ01(φ01(φ01(φ01(1))))))) = φ01(φ01(φ01(φ01(φ01(φ01(φ01(φ01(0)))))))) = ωωωωωωω φ01(φ01(φ01(φ01(φ01(φ01(φ01(φ01(1)))))))) = φ01(φ01(φ01(φ01(φ01(φ01(φ01(φ01(φ01(0))))))))) = ωωωωωωωω ... FS #6 φ03(0) = φ01(φ12(φ23(0))) = φ02(φ23(0)) = φ02(1) = φ01(φ12(1)) = φ01(Ω) = φ01(φ12(φ12(0))) = φ(1, 0) = ε0 ε0 + 1 ε0 + 2 ε0 + 3 ε0 + 4 ε0 + 5 ε0 + 6 ε0 + 7 ... FS #7 ε0 + ω ε0 + ωω ε0 + ωωω ε0 + ωωωω ε0 + ωωωωω ε0 + ωωωωωω ε0 + ωωωωωωω ... FS #8 ε02 ε03 ε04 ε05 ε06 ε07 ε08 ... FS #9 φ01(φ03(0) + 1) = ωε0 + 1 ωωε0 + 1 ωωωε0 + 1 ωωωωε0 + 1 ωωωωωε0 + 1 ωωωωωωε0 + 1 ωωωωωωωε0 + 1 ... FS #10 φ01(φ13(0) + 1) = φ01(Ω + 1) = ε1 φ01(Ω + 2) = ε2 φ01(Ω + 3) = ε3 φ01(Ω + 4) = ε4 φ01(Ω + 5) = ε5 φ01(Ω + 6) = ε6 φ01(Ω + 7) = ε7 ... FS #11 φ01(φ13(0) + φ02(0)) = φ01(Ω + ω) = εω φ01(Ω + ωω) = εωω φ01(Ω + ωωω) = εωωω φ01(Ω + ωωωω) = εωωωω φ01(Ω + ωωωωω) = εωωωωω φ01(Ω + ωωωωωω) = εωωωωωω φ01(Ω + ωωωωωωω) = εωωωωωωω ... FS #12 φ01(φ13(0) + φ03(0)) = φ01(Ω + ε0) = εε0 φ01(Ω + εε0) = εεε0 φ01(Ω + εεε0) = εεεε0 φ01(Ω + εεεε0) = εεεεε0 φ01(Ω + εεεεε0) = εεεεεε0 φ01(Ω + εεεεεε0) = εεεεεεε0 φ01(Ω + εεεεεεε0) = εεεεεεεε0 ... FS #13 φ01(φ13(0)2) = φ01(Ω2) = φ(2, 0) = ζ0 φ01(Ω3) = φ(3, 0) = η0 φ01(Ω4) = φ(4, 0) φ01(Ω5) = φ(5, 0) φ01(Ω6) = φ(6, 0) φ01(Ω7) = φ(7, 0) φ01(Ω8) = φ(8, 0) ... FS #14 φ01(φ13(0)φ02(0)) = φ01(Ωω) = φ(ω, 0) φ01(Ωωω) = φ(ωω, 0) φ01(Ωωωω) = φ(ωωω, 0) φ01(Ωωωωω) = φ(ωωωω, 0) φ01(Ωωωωωω) = φ(ωωωωω, 0) φ01(Ωωωωωωω) = φ(ωωωωωω, 0) φ01(Ωωωωωωωω) = φ(ωωωωωωω, 0) ... FS #15 φ01(φ13(0)φ03(0)) = φ01(Ωε0) = φ(ε0, 0) φ01(Ωφ01(Ωε0)) = φ(φ(ε0, 0), 0) φ01(Ωφ01(Ωφ01(Ωε0))) = φ(φ(φ(ε0, 0), 0), 0) φ01(Ωφ01(Ωφ01(Ωφ01(Ωε0)))) = φ(φ(φ(φ(ε0, 0), 0), 0), 0) φ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωε0))))) = φ(φ(φ(φ(φ(ε0, 0), 0), 0), 0), 0) φ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωε0)))))) = φ(φ(φ(φ(φ(φ(ε0, 0), 0), 0), 0), 0), 0) φ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωε0))))))) = φ(φ(φ(φ(φ(φ(φ(ε0, 0), 0), 0), 0), 0), 0), 0) ... FS #16 φ02(2) = φ01(φ12(2)) = φ01(Ω2) = φ(1, 0, 0) = Γ0 φ02(3) = φ01(φ12(3)) = φ01(Ω3) = φ(1, 0, 0, 0) φ02(4) = φ01(φ12(4)) = φ01(Ω4) = φ(1, 0, 0, 0, 0) φ02(5) = φ01(φ12(5)) = φ01(Ω5) = φ(1, 0, 0, 0, 0, 0) φ02(6) = φ01(φ12(6)) = φ01(Ω6) = φ(1, 0, 0, 0, 0, 0, 0) φ02(7) = φ01(φ12(7)) = φ01(Ω7) = φ(1, 0, 0, 0, 0, 0, 0, 0) φ02(8) = φ01(φ12(8)) = φ01(Ω8) = φ(1, 0, 0, 0, 0, 0, 0, 0, 0) ... FS #17 φ02(φ02(0)) = φ02(ω) = φ01(φ12(ω)) = φ01(Ωω) = SVO φ02(φ02(ω)) = φ01(Ωφ01(Ωω)) φ02(φ02(φ02(ω))) = φ01(Ωφ01(Ωφ01(Ωω))) φ02(φ02(φ02(φ02(ω)))) = φ01(Ωφ01(Ωφ01(Ωφ01(Ωω)))) φ02(φ02(φ02(φ02(φ02(ω))))) = φ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωω))))) φ02(φ02(φ02(φ02(φ02(φ02(ω)))))) = φ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωω)))))) φ02(φ02(φ02(φ02(φ02(φ02(φ02(ω))))))) = φ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωφ01(Ωω))))))) ... FS #18 φ02(Ω) = φ02(φ12(1)) = φ02(φ13(0)) = φ01(φ12(φ12(1))) = φ01(φ12(φ12(φ12(0)))) = φ01(ΩΩ) = LVO φ02(ΩΩ) = φ02(φ12(φ12(1))) = φ01(φ12(φ12(φ12(1)))) = φ01(φ12(φ12(φ12(φ12(0))))) = φ01(ΩΩΩ) φ02(ΩΩΩ) = φ02(φ12(φ12(φ12(1)))) = φ01(φ12(φ12(φ12(φ12(1))))) = φ01(φ12(φ12(φ12(φ12(φ12(0)))))) = φ01(ΩΩΩΩ) φ02(ΩΩΩΩ) = φ02(φ12(φ12(φ12(φ12(1))))) = φ01(φ12(φ12(φ12(φ12(φ12(1)))))) = φ01(φ12(φ12(φ12(φ12(φ12(φ12(0))))))) = φ01(ΩΩΩΩΩ) φ02(ΩΩΩΩΩ) = φ02(φ12(φ12(φ12(φ12(φ12(1)))))) = φ01(φ12(φ12(φ12(φ12(φ12(φ12(1))))))) = φ01(φ12(φ12(φ12(φ12(φ12(φ12(φ12(0)))))))) = φ01(ΩΩΩΩΩΩ) ... FS #19 φ04(0) = φ01(φ12(φ23(φ34(0)))) = φ01(φ12(φ23(1))) = φ01(φ12(Ω2)) = φ03(1) = φ02(Ω2) = BHO φ04(0) + 1 φ04(0) + 2 φ04(0) + 3 φ04(0) + 4 φ04(0) + 5 φ04(0) + 6 φ04(0) + 7 ... FS #20 φ04(0) + φ01(φ12(0)) = φ04(0) + ω φ04(0) + φ01(φ12(φ12(0))) = φ04(0) + ε0 φ04(0) + φ01(φ12(φ12(φ12(0)))) = φ04(0) + φ02(Ω) φ04(0) + φ01(φ12(φ12(φ12(φ12(0))))) = φ04(0) + φ02(ΩΩ) φ04(0) + φ01(φ12(φ12(φ12(φ12(φ12(0)))))) = φ04(0) + φ02(ΩΩΩ) φ04(0) + φ01(φ12(φ12(φ12(φ12(φ12(φ12(0))))))) = φ04(0) + φ02(ΩΩΩΩ) φ04(0) + φ01(φ12(φ12(φ12(φ12(φ12(φ12(φ12(0)))))))) = φ04(0) + φ02(ΩΩΩΩΩ) ... FS #21 φ04(0)2 φ04(0)3 φ04(0)4 φ04(0)5 φ04(0)6 φ04(0)7 ... FS #22 ωφ04(0) + 1 ωωφ04(0) + 1 ωωωφ04(0) + 1 ωωωωφ04(0) + 1 ωωωωωφ04(0) + 1 ωωωωωωφ04(0) + 1 ωωωωωωωφ04(0) + 1 ... FS #23 φ01(Ω + φ04(0) + 1) = εφ04(0) + 1 φ01(ΩΩ + φ04(0) + 1) φ01(ΩΩΩ + φ04(0) + 1) φ01(ΩΩΩΩ + φ04(0) + 1) φ01(ΩΩΩΩΩ + φ04(0) + 1) φ01(ΩΩΩΩΩΩ + φ04(0) + 1) φ01(ΩΩΩΩΩΩΩ + φ04(0) + 1) ... FS #24 φ01(φ12(Ω2) + 1) = φ01(φ13(1) + 1) = φ01(φ14(0) + 1) φ01(φ12(Ω2) + 2) = φ01(φ14(0) + 2) φ01(φ12(Ω2) + 3) = φ01(φ14(0) + 3) φ01(φ12(Ω2) + 4) = φ01(φ14(0) + 4) φ01(φ12(Ω2) + 5) = φ01(φ14(0) + 5) φ01(φ12(Ω2) + 6) = φ01(φ14(0) + 6) φ01(φ12(Ω2) + 7) = φ01(φ14(0) + 7) ... FS #25 φ01(φ12(Ω2) + ω) = φ01(φ14(0) + ω) φ01(φ12(Ω2) + ε0) = φ01(φ14(0) + ε0) φ01(φ12(Ω2) + φ02(Ω)) = φ01(φ14(0) + φ02(Ω)) φ01(φ12(Ω2) + φ02(ΩΩ)) = φ01(φ14(0) + φ02(ΩΩ)) φ01(φ12(Ω2) + φ02(ΩΩΩ)) = φ01(φ14(0) + φ02(ΩΩΩ)) φ01(φ12(Ω2) + φ02(ΩΩΩΩ)) = φ01(φ14(0) + φ02(ΩΩΩΩ)) φ01(φ12(Ω2) + φ02(ΩΩΩΩΩ)) = φ01(φ14(0) + φ02(ΩΩΩΩΩ)) ... FS #26 φ01(φ12(Ω2) + φ04(0)) = φ01(φ14(0) + φ04(0)) φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ04(0))) = φ01(φ14(0) + φ01(φ14(0) + φ04(0))) φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ04(0)))) = φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ04(0)))) φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ04(0))))) = φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ04(0))))) φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ04(0)))))) = φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ04(0)))))) φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ04(0))))))) = φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ04(0))))))) φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ01(φ12(Ω2) + φ04(0)))))))) = φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ01(φ14(0) + φ04(0)))))))) ... FS #27 φ01(φ12(Ω2) + Ω) = φ01(φ14(0) + φ12(1)) φ01(φ12(Ω2) + ΩΩ) = φ01(φ14(0) + φ12(Ω)) φ01(φ12(Ω2) + ΩΩΩ) = φ01(φ14(0) + φ12(ΩΩ)) φ01(φ12(Ω2) + ΩΩΩΩ) = φ01(φ14(0) + φ12( ΩΩΩ)) φ01(φ12(Ω2) + ΩΩΩΩΩ) = φ01(φ14(0) + φ12(ΩΩΩΩ)) φ01(φ12(Ω2) + ΩΩΩΩΩΩ) = φ01(φ14(0) + φ12(ΩΩΩΩΩ)) φ01(φ12(Ω2) + ΩΩΩΩΩΩΩ) = φ01(φ14(0) + φ12(ΩΩΩΩΩΩ)) ... FS #28 φ01(φ12(Ω2)2) = φ01(φ14(0)2) φ01(φ12(Ω2)3) = φ01(φ14(0)3) φ01(φ12(Ω2)4) = φ01(φ14(0)4) φ01(φ12(Ω2)5) = φ01(φ14(0)5) φ01(φ12(Ω2)6) = φ01(φ14(0)6) φ01(φ12(Ω2)7) = φ01(φ14(0)7) φ01(φ12(Ω2)8) = φ01(φ14(0)8) ... FS #29 φ01(φ12(Ω2)ω) = φ01(φ14(0)ω) φ01(φ12(Ω2)ε0) = φ01(φ14(0)ε0) φ01(φ12(Ω2)φ02(Ω)) = φ01(φ14(0)φ02(Ω)) φ01(φ12(Ω2)φ02(ΩΩ)) = φ01(φ14(0)φ02(ΩΩ)) φ01(φ12(Ω2)φ02(ΩΩΩ)) = φ01(φ14(0)φ02(ΩΩΩ)) φ01(φ12(Ω2)φ02(ΩΩΩΩ)) = φ01(φ14(0)φ02(ΩΩΩΩ)) φ01(φ12(Ω2)φ02(ΩΩΩΩΩ)) = φ01(φ14(0)φ02(ΩΩΩΩΩ)) ... FS #30 φ01(φ12(Ω2)φ04(0)) = φ01(φ14(0)φ04(0)) φ01(φ12(Ω2)φ01(φ12(Ω2)φ04(0))) = φ01(φ14(0)φ01(φ14(0)φ04(0))) φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ04(0)))) = φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ04(0)))) φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ04(0))))) = φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ04(0))))) φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ04(0)))))) = φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ04(0)))))) φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ04(0))))))) = φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ04(0))))))) φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ01(φ12(Ω2)φ04(0)))))))) = φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ01(φ14(0)φ04(0)))))))) ... FS #31 φ01(Ωφ12(Ω2) + 1) = φ01(φ12(φ12(Ω2) + 1)) = φ02(φ12(Ω2) + 1) φ01(ΩΩφ12(Ω2) + 1) = φ01(φ12(φ12(φ12(Ω2) + 1))) = φ02(Ωφ12(Ω2) + 1) φ01(ΩΩΩφ12(Ω2) + 1) = φ01(φ12(φ12(φ12(φ12(Ω2) + 1)))) = φ02(ΩΩφ12(Ω2) + 1) φ01(ΩΩΩΩφ12(Ω2) + 1) = φ01(φ12(φ12(φ12(φ12(φ12(Ω2) + 1))))) = φ02(ΩΩΩφ12(Ω2) + 1) φ01(ΩΩΩΩΩφ12(Ω2) + 1) = φ01(φ12(φ12(φ12(φ12(φ12(φ12(Ω2) + 1)))))) = φ02(ΩΩΩΩφ12(Ω2) + 1) φ01(ΩΩΩΩΩΩφ12(Ω2) + 1) = φ01(φ12(φ12(φ12(φ12(φ12(φ12(φ12(Ω2) + 1))))))) = φ02(ΩΩΩΩΩφ12(Ω2) + 1) φ01(ΩΩΩΩΩΩΩφ12(Ω2) + 1) = φ01(φ12(φ12(φ12(φ12(φ12(φ12(φ12(φ12(Ω2) + 1)))))))) = φ02(ΩΩΩΩΩΩφ12(Ω2) + 1) ... FS #32 φ02(Ω2 + 1) = φ02(φ23(1) + 1) = φ02(φ24(0) + 1) φ02(Ω2 + 2) = φ02(φ24(0) + 2) φ02(Ω2 + 3) = φ02(φ24(0) + 3) φ02(Ω2 + 4) = φ02(φ24(0) + 4) φ02(Ω2 + 5) = φ02(φ24(0) + 5) φ02(Ω2 + 6) = φ02(φ24(0) + 6) φ02(Ω2 + 7) = φ02(φ24(0) + 7) ... FS #33 φ02(Ω2 + ω) = φ02(φ24(0) + ω) φ02(Ω2 + ε0) = φ02(φ24(0) + ε0) φ02(Ω2 + φ02(Ω)) = φ02(φ24(0) + φ02(Ω)) φ02(Ω2 + φ02(ΩΩ)) = φ02(φ24(0) + φ02(ΩΩ)) φ02(Ω2 + φ02(ΩΩΩ)) = φ02(φ24(0) + φ02(ΩΩΩ)) φ02(Ω2 + φ02(ΩΩΩΩ)) = φ02(φ24(0) + φ02(ΩΩΩΩ)) φ02(Ω2 + φ02(ΩΩΩΩΩ)) = φ02(φ24(0) + φ02(ΩΩΩΩΩ)) ... FS #34 φ02(Ω2 + φ04(0)) = φ02(φ24(0) + φ04(0)) φ02(Ω2 + φ02(Ω2 + φ04(0))) φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ04(0)))) φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ04(0))))) φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ04(0)))))) φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ04(0))))))) φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ02(Ω2 + φ04(0)))))))) ... FS #35 φ02(Ω2 + Ω) = φ02(φ24(0) + φ13(0)) φ02(Ω2 + ΩΩ) φ02(Ω2 + ΩΩΩ) φ02(Ω2 + ΩΩΩΩ) φ02(Ω2 + ΩΩΩΩΩ) φ02(Ω2 + ΩΩΩΩΩΩ) φ02(Ω2 + ΩΩΩΩΩΩΩ) ... FS #36 φ02(Ω2 + φ12(Ω2)) = φ01(φ12(Ω2 + φ12(Ω2))) = φ02(φ24(0) + φ14(0)) φ02(Ω2 + φ12(Ω2 + φ12(Ω2))) φ02(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2)))) φ02(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2))))) φ02(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2)))))) φ02(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2))))))) φ02(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2 + φ12(Ω2)))))))) ... FS #37 φ02(Ω22) = φ02(φ24(0)2) φ02(Ω23) φ02(Ω24) φ02(Ω25) φ02(Ω26) φ02(Ω27) φ02(Ω28) ... FS #38 φ02(Ω2ω) = φ02(φ24(0)φ02(0)) φ02(Ω2ε0) = φ02(φ24(0)φ03(0)) φ02(Ω2φ02(Ω)) φ02(Ω2φ02(ΩΩ)) φ02(Ω2φ02(ΩΩΩ)) φ02(Ω2φ02(ΩΩΩΩ)) φ02(Ω2φ02(ΩΩΩΩΩ)) ... FS #39 φ02(Ω2φ04(0)) = φ02(φ24(0)φ04(0)) φ02(Ω2φ02(Ω2φ04(0))) φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))) φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0))))) φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))))) φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0))))))) φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))))))) ... FS #40 φ02(Ω2Ω) = φ02(φ24(0)φ13(0)) φ02(Ω2ΩΩ) φ02(Ω2ΩΩΩ) φ02(Ω2ΩΩΩΩ) φ02(Ω2ΩΩΩΩΩ) φ02(Ω2ΩΩΩΩΩΩ) φ02(Ω2ΩΩΩΩΩΩΩ) ... FS #41 φ02(Ω2φ12(Ω2)) = φ01(φ12(Ω2φ12(Ω2))) = φ02(φ24(0)φ14(0)) φ02(Ω2φ12(Ω2φ12(Ω2))) φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))) φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2))))) φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))))) φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2))))))) φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))))))) ... FS #42 φ03(2) = φ02(Ω22) φ03(3) = φ02(Ω23) φ03(4) = φ02(Ω24) φ03(5) = φ02(Ω25) φ03(6) = φ02(Ω26) φ03(7) = φ02(Ω27) φ03(8) = φ02(Ω28) ... FS #43 φ03(ω) = φ02(Ω2ω) φ03(ε0) = φ02(Ω2ε0) φ03(φ02(Ω)) = φ02(Ω2φ02(Ω)) φ03(φ02(ΩΩ)) = φ02(Ω2φ02(ΩΩ)) φ03(φ02(ΩΩΩ)) = φ02(Ω2φ02(ΩΩΩ)) φ03(φ02(ΩΩΩΩ)) = φ02(Ω2φ02(ΩΩΩΩ)) φ03(φ02(ΩΩΩΩΩ)) = φ02(Ω2φ02(ΩΩΩΩΩ)) ... FS #44 φ03(φ04(0)) = φ02(Ω2φ04(0)) φ03(φ02(Ω2φ04(0))) = φ02(Ω2φ02(Ω2φ04(0))) φ03(φ02(Ω2φ02(Ω2φ04(0)))) = φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))) φ03(φ02(Ω2φ02(Ω2φ02(Ω2φ04(0))))) = φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0))))) φ03(φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))))) = φ02(Ω2 φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))))) φ03(φ02(Ω2 φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0))))))) = φ02(Ω2φ02(Ω2 φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0))))))) φ03(φ02(Ω2φ02(Ω2 φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))))))) = φ02(Ω2φ02(Ω2φ02(Ω2 φ02(Ω2φ02(Ω2φ02(Ω2φ02(Ω2φ04(0)))))))) ... FS #45 φ03(Ω) = φ02(Ω2Ω) φ03(ΩΩ) = φ02(Ω2ΩΩ) φ03(ΩΩΩ) = φ02(Ω2ΩΩΩ) φ03(ΩΩΩΩ) = φ02(Ω2ΩΩΩΩ) φ03(ΩΩΩΩΩ) = φ02(Ω2ΩΩΩΩΩ) φ03(ΩΩΩΩΩΩ) = φ02(Ω2ΩΩΩΩΩΩ) φ03(ΩΩΩΩΩΩΩ) = φ02(Ω2ΩΩΩΩΩΩΩ) ... FS #46 φ03(φ12(Ω2)) = φ02(Ω2φ12(Ω2)) φ03(φ12(Ω2φ12(Ω2))) = φ02(Ω2φ12(Ω2φ12(Ω2))) φ03(φ12(Ω2φ12(Ω2φ12(Ω2)))) = φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))) φ03(φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2))))) = φ02(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2))))) φ03(φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))))) = φ02(Ω2 φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))))) φ03(φ12(Ω2 φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2))))))) = φ02(Ω2φ12(Ω2 φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2))))))) φ03(φ12(Ω2φ12(Ω2 φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))))))) = φ02(Ω2φ12(Ω2φ12(Ω2 φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2φ12(Ω2)))))))) ... FS #47 φ03(Ω2) = φ02(Ω2Ω2) φ03(Ω2Ω2) = φ02(Ω2Ω2Ω2) φ03(Ω2Ω2Ω2) = φ02(Ω2Ω2Ω2Ω2) φ03(Ω2Ω2Ω2Ω2) = φ02(Ω2Ω2Ω2Ω2Ω2) φ03(Ω2Ω2Ω2Ω2Ω2) = φ02(Ω2Ω2Ω2Ω2Ω2Ω2) φ03(Ω2Ω2Ω2Ω2Ω2Ω2) = φ02(Ω2Ω2Ω2Ω2Ω2Ω2Ω2) φ03(Ω2Ω2Ω2Ω2Ω2Ω2Ω2) = φ02(Ω2Ω2Ω2Ω2Ω2Ω2Ω2Ω2) ... FS #48 φ05(0) φ06(0) φ07(0) φ08(0) φ09(0) φ010(0) φ011(0) ... FS #49 φ0ω(0) φ0ω(0) + 1 φ0ω(0) + 2 φ0ω(0) + 3 φ0ω(0) + 4 φ0ω(0) + 5 φ0ω(0) + 6 ... FS #49 φ0ω(0) + φ02(0) = φ0ω(0) + ω φ0ω(0) + φ03(0) = φ0ω(0) + ε0 φ0ω(0) + φ04(0) φ0ω(0) + φ05(0) φ0ω(0) + φ06(0) φ0ω(0) + φ07(0) φ0ω(0) + φ08(0) ... FS #50 φ0ω(0)2 φ0ω(0)3 φ0ω(0)4 φ0ω(0)5 φ0ω(0)6 φ0ω(0)7 φ0ω(0)8 ... FS #51 ωφ0ω(0) + 1 ωωφ0ω(0) + 1 ωωωφ0ω(0) + 1 ωωωωφ0ω(0) + 1 ωωωωωφ0ω(0) + 1 ωωωωωωφ0ω(0) + 1 ωωωωωωωφ0ω(0) + 1 ...FS #52 φ01(φ13(0) + φ0ω(0) + 1) = φ01(Ω + φ0ω(0) + 1) φ01(φ14(0) + φ0ω(0) + 1) φ01(φ15(0) + φ0ω(0) + 1) φ01(φ16(0) + φ0ω(0) + 1) φ01(φ17(0) + φ0ω(0) + 1) φ01(φ18(0) + φ0ω(0) + 1) φ01(φ19(0) + φ0ω(0) + 1) ... FS #53 φ01(φ1ω(0) + 1) φ02(φ2ω(0) + 1) φ03(φ3ω(0) + 1) φ04(φ4ω(0) + 1) φ05(φ5ω(0) + 1) φ06(φ6ω(0) + 1) φ07(φ7ω(0) + 1) ... FS #54 φ0ω(1) = φ0ω(φωω + 1(0)) = φ0ω + 1(0) φ0ω(2) φ0ω(3) φ0ω(4) φ0ω(5) φ0ω(6) φ0ω(7) ... FS #55 φ0ω(φ02(0)) = φ0ω(ω) φ0ω(φ03(0)) = φ0ω(ε0) φ0ω(φ04(0)) φ0ω(φ05(0)) φ0ω(φ06(0)) φ0ω(φ07(0)) φ0ω(φ08(0)) ... FS #56 φ0ω(φ0ω(0)) φ0ω(φ0ω(φ0ω(0))) φ0ω(φ0ω(φ0ω(φ0ω(0)))) φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(0))))) φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(0)))))) φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(0))))))) φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(φ0ω(0)))))))) ... FS #57 φ0ω(Ω) φ0ω(Ω2) φ0ω(Ω3) φ0ω(Ω4) φ0ω(Ω5) φ0ω(Ω6) φ0ω(Ω7) ... FS #58 φ0ω(Ωω) = φ0ω(φωω + 1(1)) = φ0ω + 1(1) = φ0ω(φωω + 1(φω + 1ω + 2(0))) = φ0ω + 2(0) φ0ω(ΩωΩω) φ0ω(ΩωΩωΩω) φ0ω(ΩωΩωΩωΩω) φ0ω(ΩωΩωΩωΩωΩω) φ0ω(ΩωΩωΩωΩωΩωΩω) φ0ω(ΩωΩωΩωΩωΩωΩωΩω) ... FS #59 φ0ω(φωω + 1(Ωω + 1)) = φ0ω(φωω + 1(φω + 1ω + 2(1))) = φ0ω(φωω + 1(φω + 1ω + 2(φω + 2ω + 3(0)))) = φ0ω + 3(0) = TFB? φ0ω + 4(0) φ0ω + 5(0) φ0ω + 6(0) φ0ω + 7(0) φ0ω + 8(0) φ0ω + 9(0) ... FS #60 φ0ω2(0) φ0ω3(0) φ0ω4(0) φ0ω5(0) φ0ω6(0) φ0ω7(0) φ0ω8(0) ... FS #61 φ0ω2(0) φ0ω3(0) φ0ω4(0) φ0ω5(0) φ0ω6(0) φ0ω7(0) φ0ω8(0) ... FS #62 φ0ωω(0) φ0ωωω(0) φ0ωωωω(0) φ0ωωωωω(0) φ0ωωωωωω(0) φ0ωωωωωωω(0) φ0ωωωωωωωω(0) ... FS #63 φ0φ03(0)(0) = φ0ε0(0) φ0φ04(0)(0) φ0φ05(0)(0) φ0φ06(0)(0) φ0φ07(0)(0) φ0φ08(0)(0) φ0φ09(0)(0) ... FS #64 φ0φ0ω(0)(0) φ0φ0φ0ω(0)(0)(0) φ0φ0φ0φ0ω(0)(0)(0)(0) φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0) φ0φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0) φ0φ0φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0)(0) φ0φ0φ0φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0)(0)(0) ... FS #65 φ0Ω(0) φ0Ω2(0) φ0Ω3(0) φ0Ω4(0) φ0Ω5(0) φ0Ω6(0) φ0Ω7(0) ... FS #66 φ0Ωφ02(0)(0) = φ0Ωω(0) φ0Ωφ03(0)(0) = φ0Ωε0(0) φ0Ωφ04(0)(0) φ0Ωφ05(0)(0) φ0Ωφ06(0)(0) φ0Ωφ07(0)(0) φ0Ωφ08(0)(0) ... FS #67 φ0Ωφ0φ02(0)(0)(0) = φ0Ωφ0ω(0)(0) φ0Ωφ0φ03(0)(0)(0) = φ0Ωφ0ε0(0)(0) φ0Ωφ0φ04(0)(0)(0) φ0Ωφ0φ05(0)(0)(0) φ0Ωφ0φ06(0)(0)(0) φ0Ωφ0φ07(0)(0)(0) φ0Ωφ0φ08(0)(0)(0) ... FS #68 φ0Ωφ0φ0ω(0)(0)(0) φ0Ωφ0φ0φ0ω(0)(0)(0)(0) φ0Ωφ0φ0φ0φ0ω(0)(0)(0)(0)(0) φ0Ωφ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0) φ0Ωφ0φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0)(0) φ0Ωφ0φ0φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0)(0)(0) φ0Ωφ0φ0φ0φ0φ0φ0φ0φ0ω(0)(0)(0)(0)(0)(0)(0)(0)(0) ... FS #69 φ0Ωφ0Ω(0)(0) φ0Ω φ0Ωφ0Ω(0)(0)(0) φ0Ωφ0Ω φ0Ωφ0Ω(0)(0)(0)(0) φ0Ωφ0Ωφ0Ω φ0Ωφ0Ω(0)(0)(0)(0)(0) φ0Ωφ0Ωφ0Ωφ0Ω φ0Ωφ0Ω(0)(0)(0)(0)(0)(0) φ0Ωφ0Ωφ0Ωφ0Ωφ0Ω φ0Ωφ0Ω(0)(0)(0)(0)(0)(0)(0) φ0Ωφ0Ωφ0Ωφ0Ωφ0Ωφ0Ω φ0Ωφ0Ω(0)(0)(0)(0)(0)(0)(0)(0) ... FS #70 φ0ΩΩ(0) φ0ΩΩΩ(0) φ0ΩΩΩΩ(0) φ0ΩΩΩΩΩ(0) φ0ΩΩΩΩΩΩ(0) φ0ΩΩΩΩΩΩΩ(0) φ0ΩΩΩΩΩΩΩΩ(0) ... Currently I do not know, is TFB φ0ω + 3(0), because I still do not understand OCF's, and I can't check it. TFB also may be φ0ω + 2(0) or φ0ω + 1(0). Category:Blog posts